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The barrier–cluster model applied to chalcogenide glasses
Journal of Non-Crystalline Solids 353 (2007) 1920–1924 www.elsevier.com/locate/jnoncrysol
The barrier–cluster model applied to chalcogenide glasses Ivan Banik Faculty CE, Slovak University of Technology, 813 68 Bratislava, Slovak Republic Available online 2 April 2007
Abstract The aim of this article is to familiarize the readers with barrier–cluster model of the non-crystalline semiconductors, which is based on the assumption that in non-crystalline semiconductors, there exist micro-regions separated from each other by potential barriers. We suppose that these micro-regions in chalcogenide glasses are created by closed clusters. This model allows explanation not only of a number of important optical and electrical features of chalcogenide glasses, but also the results of X-ray structure measurements and ESR experiments. This concept gives a new look at the density of states within the forbidden band and at the exponential tails of the optical absorption. 2007 Elsevier B.V. All rights reserved. PACS: 71.23.Cq; 72.40.+w; 78.20.Bh; 78.40.Fy; 78.55.Qr; 78.66.Jq; 78.90.+t Keywords: Amorphous semiconductors; Optical properties; Absorption; Electroluminescence; Luminescence; Photoinduced effects; Photoconductivity; Structure; Defects
1. Introduction Much attention has been paid for the last few decades to clarification of the structure and physical properties of chalcogenide glasses. The experimental facts pile up, but the theory remains behind. There is no satisfactory model and argument goes on even on its most fundamental starting points. No model has been suggested yet that would explain sufficiently the wide range of observed phenomena. The problem has not been solved even by the most modern technical means. The first fundamental considerations of non-crystalline semiconductors are reviewed in Refs. [1,2]. Subsequent developments are described in Refs. [3– 8]. Although the chalcogenide glasses are widely used in modern techniques and technology [9–12], many of physical processes in these materials remain to be a mystery. In this paper, we will present the essence of the barrier– cluster model and some concrete examples of its possibilities by clarification of various optical phenomena, espe-
E-mail address: [email protected] 0022-3093/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.01.055
cially of optical absorption in region of the exponential tails. 2. Barrier–cluster model of non-crystalline semiconductor 2.1. Structure of chalcogenide glass In spite of tremendous effort dedicated to chalcogenide glasses, the structure and properties of these materials have not been completely understood yet [13–20]. It was pointed out many years ago that no ESR signal was detected in amorphous chalcogenides (a-Se, a-As2S3). This observation, viz. the absence of spins in chalcogenide glasses, led Anderson [13] and Street and Mott [14] to formulate their negative effective correlation energy (negativeU) models, in terms of charge defects. The model allows for the presence of dangling bonds, but with the combination of positively and negatively charged dangling bonds, D+ and D 0 having no spin- and two spin-paired electrons respectively, being energetically favored over the neutral dangling bond Do with one unpaired electron. The defectbased version of the negative-U model was subsequently
I. Banik / Journal of Non-Crystalline Solids 353 (2007) 1920–1924
developed by Kastner et al. [15] for the case of a-Se with singly coordinated, negatively charged selenium ðC01 Þ, and a threefold coordinated, positively charged site ðCþ 3Þ emerged as the most likely equilibrium defect configurations. Thus, the valence alternation pair concept appeared. Up to now, no direct experimental evidence was found for the valence alternation pair in chalcogenide glasses. Recently, Tanaka [21] has questioned the presence of the charged defects on the basis of optical absorption measurements on highly purified As2S3 samples. The discovery of the fullerenes led to the suggestion that other low-dimensional covalent systems, as e.g. chalcogenides, could create special configurations similar to fullerenes or nanotubes. Fullerene-like materials and nanotubes with or without closed ends were simulated in order to demonstrate that special ‘objects’ are possible at least in arsenic chalcogenides [22–26]. The self-organization in arsenic chalcogenides is basic to the formation of the low-dimensional objects in chalcogenides. Several types of closed nanoclusters of As2S3 were built. The fairly good agreement of several characteristics, calculated from the model, with the experimental ones allows us to conclude that a model with closed ends is very attractive and could be improved. As a consequence, the dangling bonds are naturally eliminated during the glass formation and, therefore, no significant amount of charged coordination defects is necessary to explain the glass structure. On the contrary, the formation of high numbers of VAPs was predicted during illumination and in the light saturated state of the glass. A closed cluster model for the binary arsenic–chalcogen glasses seems to be attractive for the explanation of the structural and electronic properties of non-crystalline chalcogenides. At the same time, the straight consequence of the model is the absence of coordination defects. 2.2. Barrier model The barrier model [27–44] assumes that an amorphous semiconductor consists of microscopic regions separated from each other by potential barriers. The barriers restrict the transition of low energy conduction electrons from one region to another. Such electrons behave between barriers of particular regions in a similar way as electrons in a crystal do. Potential barriers can be depicted inside the conduction and valence bands of an amorphous material, separating individual localized energy states at the edge of the band (Fig. 1). The electron levels between barriers, due to the small dimensions of the microscopic regions, exhibit a distinct discrete character. At the lower margin of the conduction band, a sub-band with carriers of low average mobility is created.
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Fig. 1. Electronic spectrum of non-crystalline semiconductors.
structural model. The cluster model gives us the best arguments for existence of the mentioned micro-regions in chalcogenides. The barrier–cluster model – thanks to the existence of the potential barriers – enables us to explain the wide range of optical and electronic phenomena in non-crystalline semiconductors [27–44]. The barrier model explains the problems with gap-states in chalcogenides, too. The cluster model [22–26] is able to clarify successfully the structural peculiarity of the chalcogenide glasses – the problems connected with results of the ESR-measurements and X-ray structural analyses. The barrier model, at the same time, is compatible with cluster structural model of chalcogenide glasses. The synthesis of both points of view on chalcogenide glasses is real and important. This is why we use the term barrier–cluster model for this ‘complex look’ at chalcogenide glasses. This model allows us to explain the fundamental structural, electronic and optical properties of chalcogenide glasses – the structural ones as e.g. electrical transport phenomena, optical phenomena such as the rise of exponential tails of optical absorption at the end of optical edge, as well as electro-absorption, photoelectric conductivity, and photoluminescence. The barrier–cluster model of glassy chalcogenides with large closed clusters allows for a simple explanation of the photoinduced phenomena [22–26], taking into account the interaction between clusters at the boundaries, assisted by light. Moreover, a bi-stable behavior of the clusters could explain the behavior of the glass under the action of various external factors (radiation, pressure, temperature). The barrier– cluster model with closed clusters for the chalcogenide glasses is able to give simple explanations for the general properties of these glasses. 3. Optical phenomena from the point of view of the barrier–cluster model 3.1. Optical absorption
2.3. Barrier–cluster model The barrier–cluster model combines both points of view on chalcogenide glasses – the barrier model and the cluster
In most crystalline solids, optical absorption is characterized by a sharp edge at the margin of the absorption band. Its position corresponds to the optical width of the
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forbidden band. However, the situation is different in the case of non-crystalline semiconductors. The absorption band near its border is smeared out and it creates a tail that extends deep into the forbidden band. Its profile is exponential as a rule. The exponential tails at higher temperatures tend to fit the Urbach’s formula. The slope of the tails changes with further temperature decrease. At lower temperatures, the slope of the tails does not change with further temperature decrease. However, a certain parallel shift towards lower absorption is observed. Optical phenomena in non-crystalline semiconductors represent a number of complex physical phenomena. One of the great puzzles to be explained is the origin of the exponential tails. 3.1.1. Optical absorption – the higher temperature range The starting point of the considerations in [27] on the basis of barrier model is an assumption that at proper conditions for a distinct absorption of light, the potential barriers in non-crystalline semiconductors occur, with phonons participating in the energy exchange. An electron in an optical transition accepts not only the energy hf of a photon but also the phonon energy Wphon (Fig. 2). Thus, the whole energy accepted is hf + Wphon, where Wphon is the energy acquired from a phonon ‘field’. The quantity hf is positively determined by the radiation wavelength, while Wphon has a statistical character. In principle, a photon can be absorbed only when the whole energy of the electron, hf + Wphon P Wtrans, is sufficient to cause a transition of the electron into the conduction band. It should be taken into account, however, that optical transitions on the energy levels lying just below the tops of barriers will dominate at higher temperatures. In this case, the probability of transition within a single localized region is small. The levels in adjacent micro-regions offer more possibilities for combination. However, they are connected with tunneling through barriers. Under these assumptions, the transitions to levels just below the barrier peaks will be more probable for two reasons. The transitions to lower levels will be restricted considerably by a small tunneling probability. The second reason rests in strong electron–phonon interaction caused by the barriers.
The number of electrons that can acquire such deficit of energy (Wpeak hf) from a phonon field, depends on temperature. The number of electron transitions when irradiating material by ‘low energy’ photons (and thus, also the coefficient of optical absorption a) and the number of such phonons with sufficient energy at temperature T is directly proportional. For the absorption coefficient it can be written hf W peak a exp ; ð1Þ 2kT hf W peak þ const ð2Þ ln a ¼ 2kT or, for a particular (constant) temperature ln a hf + const, which is a mathematical expression of an exponential tail of optical absorption [27]. However, the slope of tails is also temperature dependent. The formulae (1), (2) are of the some kind as the Urbach’s formula. They explain the temperature dependence of the slope of exponential tails at higher temperature. 3.1.2. Optical bandwidth of the forbidden band The concept of optical bandwidth E of the forbidden band in a non-crystalline semiconductor is used relatively often in publications dealing with this subject [1,2]. It is defined as the energy of photons of monochromatic light to which the conventional value of the absorption coefficient corresponds (a = 102 cm1). Experiments showed that the optical bandwidth E is strongly influenced by temperature T. The functional dependence E(T) (for example in the case of non-crystalline chalcogenide semiconductors) has most often a linear character, E = Eo + CT. The value dE/dT = C < 0 represents the temperature coefficient of the optical bandwidth of the forbidden band of a semiconductor. With chalcogenide materials, these values are most often in the range 5–7 · 104 eV K1 [1]. In literature [1], for example, the data are given: for As2Se3: 7 · 104 eV K1, for Se: 7 · 104 eV K1, for As2Se3–Sb2S3 semiconductors type: 6–7 · 104 eV K1, for As2S5: 6 · 104 eV K1, for CdGeAs2: 5 · 104 eV K1. These values are rather close to each other, which may not be entirely chance. 3.1.3. Optical bandwidth and barrier–cluster model The properties of optical width of the forbidden band can be explained well using the barrier–cluster model [27] for which the value E obtains a clear physical interpretation. It can be made clear at the same time, why the above values are so close to each other. However, the fact will be made clear as well that assuming the value E as optical width of the forbidden band is in fact illegitimate, because its value is influenced significantly by the concentration of phonons. From the condition that the absorption coefficient a should have a constant conventional value of a = 102 cm1, a requirement results under these circumstances that
Fig. 2. Optical transition with participating of phonons.
hf W peak ¼ K; kT
ð3Þ
I. Banik / Journal of Non-Crystalline Solids 353 (2007) 1920–1924
whence hf ¼ W peak þ CT ;
ð4Þ
where K is a constant, C = kK < 0 for k < 0, as in the region of the absorption tail, and hf Wpeak < 0. Under these circumstances, the value hf represents the optical bandwidth E = hf, so that E ¼ W peak þ CT :
ð5Þ
It is obvious that at these conditions, the value C = d(hf)/dT < 0 represents the temperature coefficient of the optical width of a forbidden band. The linear dependence obtained fits well with the data and dependences found experimentally. Supposing that diverse values of absorption for different chalcogenide materials are primarily due to different values of the exponential term, the constant K will have almost the same value for given materials. (This supposition is, however, the same as the assumption that coefficients A are almost identical for different chalcogenides.) The very close values of the temperature coefficient C = d(hf)/dT of these materials are a consequence of this as well and in a good agreement with the values introduced. Thus, the temperature dependence of optical width E of the forbidden band is not due to a real change of the bandwidth but due to energy of phonons taking part in an absorption event. With increasing temperature, the contribution of phonons increases as well. This is what enables absorption of light by lower energy photons. 3.1.4. Influence of pressure on the optical width of the forbidden band The monograph [1] considers the influence of hydrostatic pressure on the optical width of the forbidden band. The authors state some ‘discrepancies’ in this field of research, which in their opinion, are ‘only hard to grasp’. The matter in question is surprising consequences, which result for the optical width E of the forbidden band of a non-crystalline semiconductor from the thermodynamic relation oE oE aV oE ¼ ; ð6Þ oT P oT V vk op T where aV is the volume thermal expansion coefficient and vk is the coefficient of volume compressibility of the semiconductor. These values are defined by the relations 1 dV 1 dV aV ¼ and vk ¼ : ð7Þ V dT p V dp T The authors [1] analyze the experimental data from a thermodynamic point of view, and they come to the conclusion that the first term (dE/dT)V at the right side of Eq. (7) must have an enormously high negative value in non-crystalline semiconductors. They find this very surprising, because in crystalline materials this value is usually very small. From the point of view of a barrier model, this problem can be explained in a quite natural manner [27,28]. While in
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crystals the change of the optical width is connected to a real change of the forbidden band, which is fundamentally due to a change of interatomic distances, the situation is different for non-crystalline semiconductors. In these, according to the barrier model, the change of the width E is in the first instance caused by change in phonon concentration. 3.2. Optical absorption – low temperature range At low temperatures, only photons with sufficient energy, exceeding 2W, can be absorbed by material. The optical transition of an electron is connected with a tunneling process. The ‘skewed’ optical transition (Fig. 3) can be divided into two parts [27]: The first part is a vertical virtual energy transition within a single localized region (without tunneling); the subsequent second part represents a horizontal real energy tunneling transition in an adjacent localized region. Thus, absorption of a photon as a low temperature mechanism is connected to tunneling of electron through a potential barrier. One may remark that at lower temperatures, absorption of light in the vicinity of an optical absorption edge could mainly occur without tunneling process, i.e. within a single localized region. However, probability of such transitions is small due to the discrete character of the lowest levels, as well as to a small number of such levels in a single micro-region. Therefore, absorption associated with tunneling to adjacent regions is more probable in this case. In the case of a parabolic potential barrier (Fig. 4), the dependence of potential energy W(x) of electron on its position can be denoted as W(x) = ax2 + Wo, where the constant Wo, represents the height of the barrier from the bottom of the conduction band, and the quantity a determines ‘narrowness’ of the barrier. The probability of the tunneling can be written [27] as p exp{A Æ DW}, where ( rffiffiffiffiffiffi) p 2m A¼ ð8Þ h a is a constant depending on dimensions of the barrier, m is the mass of the electron, and DW is the energy difference between the energy level of peaks of potential barriers and the energy level on which the tunneling is occurring.
Fig. 3. Optical transition at low temperature.
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Fig. 4. To the tunneling process.
The solution of this problem, if we consider the influence of the low energy phonons, gives the exponential dependence ln a = Ahf + B(T) [27–35,37–40]. The term B(T), which depends on temperature, will cause a parallel shift of the straight lines. 4. Conclusions The present state of physics of non-crystalline solids still appears complex. The range of experimental research in this area is very extensive and calls for theoretical analysis. The most important phenomena observed in non-crystalline semiconductors can be explained on the basis of the barrier–cluster model. The barrier–cluster model provides a new explanation for the density of states within the forbidden band of a semiconductor and explains why the attempts at identification of gap-states by various optical and other methods fail. The closed cluster structure can explain the absence of an ESR signal in the case of chalcogenide glasses. The barrier–cluster model can be expected stimulate a development of new interpretations of physical phenomena in non-crystalline semiconductors, and specifically in chalcogenide glasses. Acknowledgment This research was supported by ESF and KEGA grants. References [1] N.F. Mott, E.A. Davis, Electron Processes in Non-Crystalline Materials, Clarendon, Oxford, 1979 (Elektronnyje processy v nekristalicˇeskich vesˇcˇestvach, Mir, Moskva (1982)). [2] M.H. Brodsky, Amorphous Semiconductors, Springer, Berlin, Heidelberg, New York, 1979 (Amorfnyje poluprovodniky, Mir, Moskva (1982)). [3] B.T. Kolomijec, in: Conference of Amorphous Semiconductors 78, Pardubice, 1978, p. 3. [4] M.A. Popescum, Non-Crystalline Chalcogenides, Solid State Science and Technology Library, vol. 8, Kluwer Academic, Dordrecht, Boston/London, 2000. [5] A.M. Adriesh, M.S. Iovu, Moldavian J. Phys. Sci. 2 (3&4) (2003) 246. [6] R. Faimar, B. Ushkov (Eds.), Semiconducting Chalcogenide Glass I: Glass Formation, Structure, and Stimulated Transformations Chalcogenide Glasses, Semiconductors and Semimetals, vol. 78, Elsevier– Academic, Amsterdam, Boston, London, New York, 2004.
[7] D. Lezˇal, P. Macko, Non-Crystalline Semiconductors, ALFA, Bratislava, 1988. [8] H. Overhof, J. Non-Cryst. Solids 227–230 (1998) 15. [9] Y. Hamakawa (Ed.), Amorphous Semiconductor – Technologies and Devices, North-Holland, Amsterdam, New York, Oxford, OHM, Tokyo-Osaka-Kyoto, 1983 (Amorfnyje poluprovodniky i pribory na ich osnove, ed. S.S. Gorelika, Metalurgia, Moskva, (1986)). [10] A.M. Adriesh, M.S. Iovu, S.D. Shutov, J. Optoelectron. Adv. Mater. 4 (3) (2002) 63. [11] A. Popescu. Photoinduced phenomenon and elements for integrated optics based on non-crystalline chalcogenide semiconductors, Ed.: Chisinau, 2003. [12] A.E. Kolobov (Ed.), Photoinduced Metastability in Amorphous Semiconductors, Wiley-VCH GmbH, 2003. [13] P.W. Anderson, Phys. Rev. Lett. 34 (1975) 953. [14] R.A. Street, N.F. Mott, Phys. Rev. Lett. 35 (1975) 1293. [15] M. Kastner, D. Adler, H. Fritzsche, Phys. Rev. Lett. 37 (1976) 1504. [16] G.J. Adrieanssens, A. Stesmans, J. Optoelectron. Adv. Mater. 4 (4) (2002) 837. [17] T.F. Mazec, S.K. Pavlov, M. Za´vetova´, V. Vorlı´cˇek, in: Conference of Amorphous Semiconductors 78, vol. 1, Pardubice, 1978, p. 432. [18] S.A. Kozuikhin, E.N. Voronkov, N.P. Kuz’mina, J. Non-Cryst. Solids 352 (9–20) (2006) 1547. [19] R.A. Street, T.M. Searle, I.G. Austin, R.S. Sussman, J. Phys. C: Solid State Phys. 7 (1974) 1582. ˇ ervinka, J. Non-Cryst. Solids 106 (1988) 291. [20] L. C [21] K. Tanaka, J. Optoelectron. Adv. Mater. 3 (2) (2001) 189. [22] M. Popescu, J. Optoelectron. Adv. Mater. 6 (4) (2004) 1147. [23] A. Lorinczi, F. Sava, Chalcogenide Lett. 2 (1) (2005) 1. [24] M. Popescu, Conference of Amorphous Semiconductors 78, vol. 1, Pardubice, 1978, p. 185. [25] S. Zamfira, M. Popescu, F. Sava, Chalcogenide Lett. 2 (6) (2005) 55. [26] M. Popescu, F. Sava, A. Lorinczi, Chalcogenide Lett. 1 (9) (2004) 111. [27] I. Banik, CEJP 3 (2) (2005) 270. [28] I. Banik, CEJP, submitted for publication. [29] I. Banik, in: Proceedings of the 15th Conference of Slovak and Czech. Physicists, Kosˇice, 2006, p. 45. [30] I. Banik, in: Proceedings of the Nineth International Workshop APCOM 2003, Mala´ Lucˇivna´, 2003, p. 32. [31] I. Banik, in: Proceedings of the Conference TVV 2006, Olomouc, Votobia Prag, 2006, p. 361. [32] I. Banik, in: Proceedings of the Conference TTV 2005, Olomouc, Votobia Prag, 2005, p. 414. [33] I. Banik, in: Proceedings of the Conference Research on UT SR, Nitra, SUA Nitra, 2005, p. 14. [34] I. Banı´k, in: Conference on Physical and Material Engineering, Richnˇava, 2003, p. 179. [35] I. Banı´k, Didmattech 03, VOTOBIA, Olomouc, 2003, p. 88. [36] I. Banı´k, Czech. J. Phys. B 24 (1974) 1114. [37] I. Banik, in: Proceedings of the FCE SUT, SUT, Bratislava, 1983, p. 221. [38] I. Banı´k, in: Proceedings of the Amorphous Semiconductors 78, Pardubice, 1978, p. 529. [39] I. Banı´k, in: Proceedings of the Seventh Conference Czech–Slov. Physicist, Prague, 1981, p. 5. [40] I. Banı´k, in: Proceedings of the Fifth Conference Czech. Phys., Kosˇice, 1977, p. 643. [41] I. Banik, The transport properties of amorphous semiconductors in high electric fields (in slovak), SVSˇT-SUT, Bratislava, 1975. [42] I. Banı´k, Czech. J. Phys. B 25 (1975) 552. ˇ s. cˇas. fyz. A 26 (1976) 496. [43] I. Banı´k, C [44] I. Banı´k, Electrical and optical properties of non-crystalline semiconductors under normal and high pressure, SUT Bratislava, 2002.
The barrier–cluster model applied to chalcogenide glasses Ivan Banik Faculty CE, Slovak University of Technology, 813 68 Bratislava, Slovak Republic Available online 2 April 2007
Abstract The aim of this article is to familiarize the readers with barrier–cluster model of the non-crystalline semiconductors, which is based on the assumption that in non-crystalline semiconductors, there exist micro-regions separated from each other by potential barriers. We suppose that these micro-regions in chalcogenide glasses are created by closed clusters. This model allows explanation not only of a number of important optical and electrical features of chalcogenide glasses, but also the results of X-ray structure measurements and ESR experiments. This concept gives a new look at the density of states within the forbidden band and at the exponential tails of the optical absorption. 2007 Elsevier B.V. All rights reserved. PACS: 71.23.Cq; 72.40.+w; 78.20.Bh; 78.40.Fy; 78.55.Qr; 78.66.Jq; 78.90.+t Keywords: Amorphous semiconductors; Optical properties; Absorption; Electroluminescence; Luminescence; Photoinduced effects; Photoconductivity; Structure; Defects
1. Introduction Much attention has been paid for the last few decades to clarification of the structure and physical properties of chalcogenide glasses. The experimental facts pile up, but the theory remains behind. There is no satisfactory model and argument goes on even on its most fundamental starting points. No model has been suggested yet that would explain sufficiently the wide range of observed phenomena. The problem has not been solved even by the most modern technical means. The first fundamental considerations of non-crystalline semiconductors are reviewed in Refs. [1,2]. Subsequent developments are described in Refs. [3– 8]. Although the chalcogenide glasses are widely used in modern techniques and technology [9–12], many of physical processes in these materials remain to be a mystery. In this paper, we will present the essence of the barrier– cluster model and some concrete examples of its possibilities by clarification of various optical phenomena, espe-
E-mail address: [email protected] 0022-3093/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.01.055
cially of optical absorption in region of the exponential tails. 2. Barrier–cluster model of non-crystalline semiconductor 2.1. Structure of chalcogenide glass In spite of tremendous effort dedicated to chalcogenide glasses, the structure and properties of these materials have not been completely understood yet [13–20]. It was pointed out many years ago that no ESR signal was detected in amorphous chalcogenides (a-Se, a-As2S3). This observation, viz. the absence of spins in chalcogenide glasses, led Anderson [13] and Street and Mott [14] to formulate their negative effective correlation energy (negativeU) models, in terms of charge defects. The model allows for the presence of dangling bonds, but with the combination of positively and negatively charged dangling bonds, D+ and D 0 having no spin- and two spin-paired electrons respectively, being energetically favored over the neutral dangling bond Do with one unpaired electron. The defectbased version of the negative-U model was subsequently
I. Banik / Journal of Non-Crystalline Solids 353 (2007) 1920–1924
developed by Kastner et al. [15] for the case of a-Se with singly coordinated, negatively charged selenium ðC01 Þ, and a threefold coordinated, positively charged site ðCþ 3Þ emerged as the most likely equilibrium defect configurations. Thus, the valence alternation pair concept appeared. Up to now, no direct experimental evidence was found for the valence alternation pair in chalcogenide glasses. Recently, Tanaka [21] has questioned the presence of the charged defects on the basis of optical absorption measurements on highly purified As2S3 samples. The discovery of the fullerenes led to the suggestion that other low-dimensional covalent systems, as e.g. chalcogenides, could create special configurations similar to fullerenes or nanotubes. Fullerene-like materials and nanotubes with or without closed ends were simulated in order to demonstrate that special ‘objects’ are possible at least in arsenic chalcogenides [22–26]. The self-organization in arsenic chalcogenides is basic to the formation of the low-dimensional objects in chalcogenides. Several types of closed nanoclusters of As2S3 were built. The fairly good agreement of several characteristics, calculated from the model, with the experimental ones allows us to conclude that a model with closed ends is very attractive and could be improved. As a consequence, the dangling bonds are naturally eliminated during the glass formation and, therefore, no significant amount of charged coordination defects is necessary to explain the glass structure. On the contrary, the formation of high numbers of VAPs was predicted during illumination and in the light saturated state of the glass. A closed cluster model for the binary arsenic–chalcogen glasses seems to be attractive for the explanation of the structural and electronic properties of non-crystalline chalcogenides. At the same time, the straight consequence of the model is the absence of coordination defects. 2.2. Barrier model The barrier model [27–44] assumes that an amorphous semiconductor consists of microscopic regions separated from each other by potential barriers. The barriers restrict the transition of low energy conduction electrons from one region to another. Such electrons behave between barriers of particular regions in a similar way as electrons in a crystal do. Potential barriers can be depicted inside the conduction and valence bands of an amorphous material, separating individual localized energy states at the edge of the band (Fig. 1). The electron levels between barriers, due to the small dimensions of the microscopic regions, exhibit a distinct discrete character. At the lower margin of the conduction band, a sub-band with carriers of low average mobility is created.
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Fig. 1. Electronic spectrum of non-crystalline semiconductors.
structural model. The cluster model gives us the best arguments for existence of the mentioned micro-regions in chalcogenides. The barrier–cluster model – thanks to the existence of the potential barriers – enables us to explain the wide range of optical and electronic phenomena in non-crystalline semiconductors [27–44]. The barrier model explains the problems with gap-states in chalcogenides, too. The cluster model [22–26] is able to clarify successfully the structural peculiarity of the chalcogenide glasses – the problems connected with results of the ESR-measurements and X-ray structural analyses. The barrier model, at the same time, is compatible with cluster structural model of chalcogenide glasses. The synthesis of both points of view on chalcogenide glasses is real and important. This is why we use the term barrier–cluster model for this ‘complex look’ at chalcogenide glasses. This model allows us to explain the fundamental structural, electronic and optical properties of chalcogenide glasses – the structural ones as e.g. electrical transport phenomena, optical phenomena such as the rise of exponential tails of optical absorption at the end of optical edge, as well as electro-absorption, photoelectric conductivity, and photoluminescence. The barrier–cluster model of glassy chalcogenides with large closed clusters allows for a simple explanation of the photoinduced phenomena [22–26], taking into account the interaction between clusters at the boundaries, assisted by light. Moreover, a bi-stable behavior of the clusters could explain the behavior of the glass under the action of various external factors (radiation, pressure, temperature). The barrier– cluster model with closed clusters for the chalcogenide glasses is able to give simple explanations for the general properties of these glasses. 3. Optical phenomena from the point of view of the barrier–cluster model 3.1. Optical absorption
2.3. Barrier–cluster model The barrier–cluster model combines both points of view on chalcogenide glasses – the barrier model and the cluster
In most crystalline solids, optical absorption is characterized by a sharp edge at the margin of the absorption band. Its position corresponds to the optical width of the
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forbidden band. However, the situation is different in the case of non-crystalline semiconductors. The absorption band near its border is smeared out and it creates a tail that extends deep into the forbidden band. Its profile is exponential as a rule. The exponential tails at higher temperatures tend to fit the Urbach’s formula. The slope of the tails changes with further temperature decrease. At lower temperatures, the slope of the tails does not change with further temperature decrease. However, a certain parallel shift towards lower absorption is observed. Optical phenomena in non-crystalline semiconductors represent a number of complex physical phenomena. One of the great puzzles to be explained is the origin of the exponential tails. 3.1.1. Optical absorption – the higher temperature range The starting point of the considerations in [27] on the basis of barrier model is an assumption that at proper conditions for a distinct absorption of light, the potential barriers in non-crystalline semiconductors occur, with phonons participating in the energy exchange. An electron in an optical transition accepts not only the energy hf of a photon but also the phonon energy Wphon (Fig. 2). Thus, the whole energy accepted is hf + Wphon, where Wphon is the energy acquired from a phonon ‘field’. The quantity hf is positively determined by the radiation wavelength, while Wphon has a statistical character. In principle, a photon can be absorbed only when the whole energy of the electron, hf + Wphon P Wtrans, is sufficient to cause a transition of the electron into the conduction band. It should be taken into account, however, that optical transitions on the energy levels lying just below the tops of barriers will dominate at higher temperatures. In this case, the probability of transition within a single localized region is small. The levels in adjacent micro-regions offer more possibilities for combination. However, they are connected with tunneling through barriers. Under these assumptions, the transitions to levels just below the barrier peaks will be more probable for two reasons. The transitions to lower levels will be restricted considerably by a small tunneling probability. The second reason rests in strong electron–phonon interaction caused by the barriers.
The number of electrons that can acquire such deficit of energy (Wpeak hf) from a phonon field, depends on temperature. The number of electron transitions when irradiating material by ‘low energy’ photons (and thus, also the coefficient of optical absorption a) and the number of such phonons with sufficient energy at temperature T is directly proportional. For the absorption coefficient it can be written hf W peak a exp ; ð1Þ 2kT hf W peak þ const ð2Þ ln a ¼ 2kT or, for a particular (constant) temperature ln a hf + const, which is a mathematical expression of an exponential tail of optical absorption [27]. However, the slope of tails is also temperature dependent. The formulae (1), (2) are of the some kind as the Urbach’s formula. They explain the temperature dependence of the slope of exponential tails at higher temperature. 3.1.2. Optical bandwidth of the forbidden band The concept of optical bandwidth E of the forbidden band in a non-crystalline semiconductor is used relatively often in publications dealing with this subject [1,2]. It is defined as the energy of photons of monochromatic light to which the conventional value of the absorption coefficient corresponds (a = 102 cm1). Experiments showed that the optical bandwidth E is strongly influenced by temperature T. The functional dependence E(T) (for example in the case of non-crystalline chalcogenide semiconductors) has most often a linear character, E = Eo + CT. The value dE/dT = C < 0 represents the temperature coefficient of the optical bandwidth of the forbidden band of a semiconductor. With chalcogenide materials, these values are most often in the range 5–7 · 104 eV K1 [1]. In literature [1], for example, the data are given: for As2Se3: 7 · 104 eV K1, for Se: 7 · 104 eV K1, for As2Se3–Sb2S3 semiconductors type: 6–7 · 104 eV K1, for As2S5: 6 · 104 eV K1, for CdGeAs2: 5 · 104 eV K1. These values are rather close to each other, which may not be entirely chance. 3.1.3. Optical bandwidth and barrier–cluster model The properties of optical width of the forbidden band can be explained well using the barrier–cluster model [27] for which the value E obtains a clear physical interpretation. It can be made clear at the same time, why the above values are so close to each other. However, the fact will be made clear as well that assuming the value E as optical width of the forbidden band is in fact illegitimate, because its value is influenced significantly by the concentration of phonons. From the condition that the absorption coefficient a should have a constant conventional value of a = 102 cm1, a requirement results under these circumstances that
Fig. 2. Optical transition with participating of phonons.
hf W peak ¼ K; kT
ð3Þ
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whence hf ¼ W peak þ CT ;
ð4Þ
where K is a constant, C = kK < 0 for k < 0, as in the region of the absorption tail, and hf Wpeak < 0. Under these circumstances, the value hf represents the optical bandwidth E = hf, so that E ¼ W peak þ CT :
ð5Þ
It is obvious that at these conditions, the value C = d(hf)/dT < 0 represents the temperature coefficient of the optical width of a forbidden band. The linear dependence obtained fits well with the data and dependences found experimentally. Supposing that diverse values of absorption for different chalcogenide materials are primarily due to different values of the exponential term, the constant K will have almost the same value for given materials. (This supposition is, however, the same as the assumption that coefficients A are almost identical for different chalcogenides.) The very close values of the temperature coefficient C = d(hf)/dT of these materials are a consequence of this as well and in a good agreement with the values introduced. Thus, the temperature dependence of optical width E of the forbidden band is not due to a real change of the bandwidth but due to energy of phonons taking part in an absorption event. With increasing temperature, the contribution of phonons increases as well. This is what enables absorption of light by lower energy photons. 3.1.4. Influence of pressure on the optical width of the forbidden band The monograph [1] considers the influence of hydrostatic pressure on the optical width of the forbidden band. The authors state some ‘discrepancies’ in this field of research, which in their opinion, are ‘only hard to grasp’. The matter in question is surprising consequences, which result for the optical width E of the forbidden band of a non-crystalline semiconductor from the thermodynamic relation oE oE aV oE ¼ ; ð6Þ oT P oT V vk op T where aV is the volume thermal expansion coefficient and vk is the coefficient of volume compressibility of the semiconductor. These values are defined by the relations 1 dV 1 dV aV ¼ and vk ¼ : ð7Þ V dT p V dp T The authors [1] analyze the experimental data from a thermodynamic point of view, and they come to the conclusion that the first term (dE/dT)V at the right side of Eq. (7) must have an enormously high negative value in non-crystalline semiconductors. They find this very surprising, because in crystalline materials this value is usually very small. From the point of view of a barrier model, this problem can be explained in a quite natural manner [27,28]. While in
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crystals the change of the optical width is connected to a real change of the forbidden band, which is fundamentally due to a change of interatomic distances, the situation is different for non-crystalline semiconductors. In these, according to the barrier model, the change of the width E is in the first instance caused by change in phonon concentration. 3.2. Optical absorption – low temperature range At low temperatures, only photons with sufficient energy, exceeding 2W, can be absorbed by material. The optical transition of an electron is connected with a tunneling process. The ‘skewed’ optical transition (Fig. 3) can be divided into two parts [27]: The first part is a vertical virtual energy transition within a single localized region (without tunneling); the subsequent second part represents a horizontal real energy tunneling transition in an adjacent localized region. Thus, absorption of a photon as a low temperature mechanism is connected to tunneling of electron through a potential barrier. One may remark that at lower temperatures, absorption of light in the vicinity of an optical absorption edge could mainly occur without tunneling process, i.e. within a single localized region. However, probability of such transitions is small due to the discrete character of the lowest levels, as well as to a small number of such levels in a single micro-region. Therefore, absorption associated with tunneling to adjacent regions is more probable in this case. In the case of a parabolic potential barrier (Fig. 4), the dependence of potential energy W(x) of electron on its position can be denoted as W(x) = ax2 + Wo, where the constant Wo, represents the height of the barrier from the bottom of the conduction band, and the quantity a determines ‘narrowness’ of the barrier. The probability of the tunneling can be written [27] as p exp{A Æ DW}, where ( rffiffiffiffiffiffi) p 2m A¼ ð8Þ h a is a constant depending on dimensions of the barrier, m is the mass of the electron, and DW is the energy difference between the energy level of peaks of potential barriers and the energy level on which the tunneling is occurring.
Fig. 3. Optical transition at low temperature.
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Fig. 4. To the tunneling process.
The solution of this problem, if we consider the influence of the low energy phonons, gives the exponential dependence ln a = Ahf + B(T) [27–35,37–40]. The term B(T), which depends on temperature, will cause a parallel shift of the straight lines. 4. Conclusions The present state of physics of non-crystalline solids still appears complex. The range of experimental research in this area is very extensive and calls for theoretical analysis. The most important phenomena observed in non-crystalline semiconductors can be explained on the basis of the barrier–cluster model. The barrier–cluster model provides a new explanation for the density of states within the forbidden band of a semiconductor and explains why the attempts at identification of gap-states by various optical and other methods fail. The closed cluster structure can explain the absence of an ESR signal in the case of chalcogenide glasses. The barrier–cluster model can be expected stimulate a development of new interpretations of physical phenomena in non-crystalline semiconductors, and specifically in chalcogenide glasses. Acknowledgment This research was supported by ESF and KEGA grants. References [1] N.F. Mott, E.A. Davis, Electron Processes in Non-Crystalline Materials, Clarendon, Oxford, 1979 (Elektronnyje processy v nekristalicˇeskich vesˇcˇestvach, Mir, Moskva (1982)). [2] M.H. Brodsky, Amorphous Semiconductors, Springer, Berlin, Heidelberg, New York, 1979 (Amorfnyje poluprovodniky, Mir, Moskva (1982)). [3] B.T. Kolomijec, in: Conference of Amorphous Semiconductors 78, Pardubice, 1978, p. 3. [4] M.A. Popescum, Non-Crystalline Chalcogenides, Solid State Science and Technology Library, vol. 8, Kluwer Academic, Dordrecht, Boston/London, 2000. [5] A.M. Adriesh, M.S. Iovu, Moldavian J. Phys. Sci. 2 (3&4) (2003) 246. [6] R. Faimar, B. Ushkov (Eds.), Semiconducting Chalcogenide Glass I: Glass Formation, Structure, and Stimulated Transformations Chalcogenide Glasses, Semiconductors and Semimetals, vol. 78, Elsevier– Academic, Amsterdam, Boston, London, New York, 2004.
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